Generalized geometry [1]}, [2]} is a branch of differential geometry
that unifies symplectic and complex geometry and
in which, in particular, vector fields and 1-forms on a manifold \(M\)  are treated on an equal footing,
as sections of a “doubled” or generalized tangent bundle, \(\mathbb {T}M=TM\oplus T^{*}M\) .
The generalization to \(\mathbb {T}M\)  of the differential structure encoded in the Lie bracket is captured by the notion of
Courant algebroids, which involve the so-called Dorfman bracket (or its antisymmetric version, the Courant bracket).
During the last years these structures have appeared naturally in theoretical high-energy physics,
as a central tool for the understanding of the geometry of string theory,
related in particular to the geometrical interpretation of T-duality (e.g. [3]}).
A prominent line of developments in this respect is Double Field Theory (see below),
where it has recently been shown [4]}, [5]}, [6]}, [7]}
that also para-Hermitian geometry is particularly well-suited to the understanding of its mathematical structure.
In this work we show that doubled geometry is also natural in general relativity,
where it provides a new perspective on `integrability issues' in special spacetimes
and their connection with twistor theory.
